Rationality and specialization

  • Yuri Tschinkel


I discuss recent advances in the study of rationality properties of algebraic varieties, with an emphasis on the specialization method initiated by Voisin.


Rationality Galois cohomology Specialization 

Mathematics Subject Classification

14E08 14J60 14M20 



I am very grateful to Fedor Bogomolov, Ivan Cheltsov, Brendan Hassett, Andrew Kresch, and Alena Pirutka for discussions on related topics. The author was partially supported by NSF grant 1601912 and by the Laboratory of Mirror Symmetry NRU HSE, RF grant 14.641.31.0001.


  1. 1.
    Abramovich, D., Karu, K., Matsuki, K., Włodarczyk, J.: Torification and factorization of birational maps. J. Am. Math. Soc. 15(3), 531–572 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. 3(25), 75–95 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auel, A., Bernardara, M.: Cycles, derived categories, and rationality. In: Surveys on Recent Developments in Algebraic Geometry, Proc. Sympos. Pure Math., vol. 95, pp. 199–266. American Mathematical Society, Providence (2017)Google Scholar
  4. 4.
    Beauville, A.: Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. École Norm. Sup. (4) 10(3), 309–391 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beauville, A.: The Lüroth problem. In: Rationality Problems in Algebraic Geometry, Lecture Notes in Mathematics, vol. 2172. Springer, Cham; Fondazione C.I.M.E., Florence (2016). arXiv:1507.02476
  6. 6.
    Beauville, A., Colliot-Thélène, J.-L., Sansuc, J.-J., Swinnerton-Dyer, P.: Variétés stablement rationnelles non rationnelles. Ann. Math. (2) 121(2), 283–318 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bloch, S., Srinivas, V.: Remarks on correspondences and algebraic cycles. Am. J. Math. 105(5), 1235–1253 (1983)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bogomolov, F., Tschinkel, Yu.: Rational curves and points on \(K3\) surfaces. Am. J. Math. 127(4), 825–835 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cassels, J.W.S., Guy, M.J.T.: On the Hasse principle for cubic surfaces. Mathematika 13, 111–120 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Clemens, C.H., Griffiths, Ph.A: The intermediate Jacobian of the cubic threefold. Ann. Math. 2(95), 281–356 (1972)Google Scholar
  11. 11.
    Clemens, C.H.: Applications of the theory of Prym varieties. In: Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), vol. 1, pp. 415–421. Canad. Math. Congress, Montreal (1975)Google Scholar
  12. 12.
    Colliot-Thélène, J.-L.: Introduction to Work of Hassett–Pirutka–Tschinkel and Schreieder (2018). arXiv:1806.00598
  13. 13.
    Colliot-Thélène, J.-L.: Non rationalité stable sur les corps quelconques (2018). arXiv:1806.00606
  14. 14.
    Colliot-Thélène, J.-L., Pirutka, A.: Hypersurfaces quartiques de dimension 3: non-rationalité stable. Ann. Sci. Éc. Norm. Supér. (4) 49(2), 371–397 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Colliot-Thélène, J.-L., Kanevsky, D., Sansuc, J.-J.: Arithmétique des surfaces cubiques diagonales. In: Diophantine Approximation and Transcendence Theory (Bonn, 1985), Lecture Notes in Mathematics, vol. 1290, pp. 1–108. Springer, Berlin (1987)Google Scholar
  16. 16.
    de Fernex, T.: Birationally rigid hypersurfaces. Invent. Math. 192(3), 533–566 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    de Fernex, T.: Fano hypersurfaces and their birational geometry. In: Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat., vol. 79, pp. 103–120. Springer, Cham (2014)Google Scholar
  18. 18.
    de Fernex, T.: Erratum to: Birationally rigid hypersurfaces [MR3049929]. Invent. Math. 203(2), 675–680 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    de Fernex, T., Fusi, D.: Rationality in families of threefolds. Rend. Circ. Mat. Palermo (2) 62(1), 127–135 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    de Fernex, T., Hacon, Chr.D.: Rigidity properties of Fano varieties. In: Current Developments in Algebraic Geometry, Math. Sci. Res. Inst. Publ., vol. 59, pp. 113–127. Cambridge University Press, Cambridge (2012)Google Scholar
  21. 21.
    Florence, M., Reichstein, Z.: The rationality problem for forms of \({\overline{M}}_{0, n}\). Bull. Lond. Math. Soc. 50(1), 148–158 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hassett, B.: Cubic fourfolds, K3 surfaces, and rationality questions. In: Rationality Problems in Algebraic Geometry, Lecture Notes in Mathematics, vol. 2172, pp. 29–66. Springer, Cham (2016)Google Scholar
  23. 23.
    Hassett, B., Tschinkel, Yu.: On stable rationality of Fano threefolds and del Pezzo fibrations (2016). arXiv:1601.07074
  24. 24.
    Hassett, B., Kresch, A., Tschinkel, Yu.: On the moduli of degree \(4\) Del Pezzo surfaces. In: Development of Moduli Theory, Advanced Studies in Pure Mathematics, vol. 69, pp. 349–386. Mathematical Society of Japan, Tokyo (2016)Google Scholar
  25. 25.
    Hassett, B., Kresch, A., Tschinkel, Yu.: Stable rationality and conic bundles. Math. Ann. 365(3–4), 1201–1217 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hassett, B., Pirutka, A., Tschinkel, Yu.: Stable rationality of quadric surface bundles over surfaces (2016). arXiv:1603.09262
  27. 27.
    Hassett, B., Pirutka, A., Tschinkel, Yu.: A very general quartic double fourfold is not stably rational (2016). arXiv:1605.03220
  28. 28.
    Hassett, B., Pirutka, A., Tschinkel, Yu.: Intersections of three quadrics in \({\bf P}^7\) (2017). arXiv:1706.01371
  29. 29.
    Hassett, B., Kresch, A., Tschinkel, Yu.: Stable rationality in smooth families of threefolds (2018). arXiv:1802.06107
  30. 30.
    Iskovskih, V.A., Manin, Yu.I.: Three-dimensional quartics and counterexamples to the Lüroth problem. Mat. Sb. (N.S.) 86(128), 140–166 (1971)Google Scholar
  31. 31.
    Kollár, J., Mella, M.: Quadratic families of elliptic curves and unirationality of degree 1 conic bundles. Am. J. Math. 139(4), 915–936 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kontsevich, M., Tschinkel, Yu.: Specialization of birational types (2017). arXiv:1708.05699
  33. 33.
    Kresch, A., Tschinkel, Yu.: Models of Brauer–Severi surface bundles (2017). arXiv:1708.06277
  34. 34.
    Kresch, A., Tschinkel, Yu.: Stable rationality of Brauer–Severi surface bundles (2017). arXiv:1709.10151
  35. 35.
    Krylov, I., Okada, T.: Stable rationality of del Pezzo fibrations of low degree over projective spaces (2017). arXiv:1701.08372
  36. 36.
    Kunyavskiĭ, B.È.: Three-dimensional algebraic tori. In: Investigations in Number Theory (Russian), pp. 90–111. Saratov. Gos. Univ., Saratov, 1987. Translated in Selecta Math. Soviet., vol. 9, no. 1, pp. 1–21 (1990)Google Scholar
  37. 37.
    Larsen, M., Lunts, V.: Motivic measures and stable birational geometry. Moscow Math. J. 3(1), 85–95 (2003)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Manin, Yu.I.: Cubic forms, North-Holland Mathematical Library, vol. 4, 2nd edn. North-Holland Publishing Co., Amsterdam (1986). Algebra, geometry, arithmetic, Translated from the Russian by M. HazewinkelGoogle Scholar
  39. 39.
    Manin, Yu.I., Tsfasman, M.A.: Rational varieties: algebra, geometry, arithmetic. Uspekhi Mat. Nauk 41(2(248)), 43–94 (1986)Google Scholar
  40. 40.
    Mella, M.: On the unirationality of 3-fold conic bundles (2014). arXiv:1403.7055
  41. 41.
    Nicaise, J., Shinder, E.: The motivic nearby fiber and degeneration of stable rationality (2017). arXiv:1708.02790
  42. 42.
    Peyre, E.: Progrès en irrationalité [d’après C. Voisin, J.-L. Colliot-thélène, B. Hassett, A. Kresch, A. Pirutka, Y. Tschinkel et al.]. Séminaire N. Bourbaki (2016)Google Scholar
  43. 43.
    Pirutka, A.: Varieties that are not stably rational, zero-cycles and unramied cohomology. In: Algebraic Geometry: Salt Lake City 2015, Proceedings of Symposia in Pure Mathematics, vol. 97, pp. 459 –484. AMS (2018)Google Scholar
  44. 44.
    Prokhorov, Yu.G.: On birational involutions of \(\mathbb{P}^3\). Izv. Ross. Akad. Nauk Ser. Mat. 77(3), 199–222 (2013)Google Scholar
  45. 45.
    Prokhorov, Yu.G.: The rationality problem for conic bundles. Uspekhi Mat. Nauk 73(3(441)), 3–88 (2018)Google Scholar
  46. 46.
    Sarkisov, V.G.: On conic bundle structures. Izv. Akad. Nauk SSSR Ser. Mat. 46(2), 371–408, 432 (1982)Google Scholar
  47. 47.
    Schreieder, S.: On the rationality problem for quadric bundles (2017). arXiv:1706.01356
  48. 48.
    Schreieder, S.: Quadric surface bundles over surfaces and stable rationality. Algebra Number Theory 12(2), 479–490 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Serre, J.-P.: Spécialisation des éléments de \({\rm Br}_2({ Q}(T_1,\ldots, T_n))\). C. R. Acad. Sci. Paris Sér. I Math. 311(7), 397–402 (1990)MathSciNetGoogle Scholar
  50. 50.
    Shokurov, V.V.: Prym varieties: theory and applications. Izv. Akad. Nauk SSSR Ser. Mat. 47(4), 785–855 (1983)MathSciNetGoogle Scholar
  51. 51.
    Tschinkel, Yu., Yang, K.: Potentially stably rational del Pezzo surfaces over nonclosed fields (2018) (preprint) Google Scholar
  52. 52.
    Voisin, C.: Unirational threefolds with no universal codimension \(2\) cycle. Invent. Math. 201(1), 207–237 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Voisin, C.: Stable birational invariants and the Lüroth problem. In: Surveys in Differential Geometry 2016. Advances in Geometry and Mathematical Physics, Surv. Differ. Geom., vol. 21, pp. 313–342. Int. Press, Somerville (2016)Google Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Simons FoundationNew YorkUSA

Personalised recommendations